Methods of scanning ions out of quadrupole ion traps for external detection are generally derived from the Mathieu parameters au and qu, which describe the stability of ions in quadrupolar fields with dimensions u. For the linear ion trap with quadrupole potentials in x and y,qx=−qy=8zeV0-p/Ω2(x02+y02)m  (1)ax=−ay=16zeU/Ω2(x02+y02)m  (2)where z is the integer charge of the ion, e is the elementary charge, U is the DC potential between the rods, V0-p is the zero-to-peak amplitude of the quadrupolar radiofrequency (rf) trapping potential, Ω is the angular rf frequency, x0 and y0 are the half distances between the rods in those respective dimensions, and m is the mass of the ion. When the dimensions in x and y are identical (x0=y0), 2r02 can be substituted for (x02+y02). Solving for m/z, the following is obtained:m/z=4V0-p/qxΩ2r02  (3)m/z=8U/axΩ2r02  (4)
Ion traps are generally operated without DC potentials (au=U=0) so that all ions occupy the q axis of the Mathieu stability diagram. In the boundary ejection method, first demonstrated in the 3D trap and in the linear ion trap, the rf amplitude is increased so that ions are ejected when their trajectories become unstable at q=0.908, giving a mass spectrum, i.e. a plot of intensity vs m/z since m/z and rf amplitude (i.e. time) are linearly related.
Resonance ejection is a similar method that improves both resolution and sensitivity. A small supplementary AC signal is applied in a dipolar manner across trapping electrodes in order to generate a small dipolar field that oscillates at the applied frequency. When this frequency, generally set near qu=0.88, matches the secular frequency of an ion in the trap, the ion will be excited or ejected from the trap depending on waveform amplitude and time of application. When the trapping rf amplitude is ramped, all ion secular frequencies increase, eventually coming into resonance with the weak dipolar field and causing their ejection in order of increasing m/z. Although a reverse scan can also be performed, the resolution and sensitivity generally suffer because of position-dependent ion frequency shifts which are observed with non-zero even higher-order field contributions (e. g. octopole).
Other variants of resonance ejection are double and triple resonance ejection, in which one or two AC frequencies are applied at nonlinear (hexapole or octopole) resonance points. These scans have been shown to greatly increase resolution and sensitivity in both conventional and miniature instruments. Rhombic ion ejection makes use of multiple frequencies in different directions for reduced space charge effects since ions being ejected will oscillate around the main ion cloud rather than pass through it. Multiple frequencies can also correspond to different ejection points, as in a compressive mass spectrometry scan, which requires acquisition of multiple scans and an algorithm to reconstruct the mass spectrum.
The radius of the trap can theoretically be scanned, but this has not been demonstrated. Instead, a more useful application is an array of traps of different radii for mass selective trapping.
An uncommon method of scanning an ion trap is to scan the main trapping rf frequency. Although useful for the analysis of microparticles and other high mass ions since lowering the rf frequency increases the mass range obtainable with a given rf amplitude maximum, calibration is difficult due to the nonlinear relationship between m/z and rf frequency. In addition, many systems which use LC tank circuits are unable to scan the rf frequency while maintaining the resonance of the circuit. Nonetheless, digital ion traps are better suited to frequency scans since they can easily modulate the period of the driving rf while providing linear calibration with an appropriate nonlinear frequency sweep.